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NONLINEAR COINTEGRATING REGRESSION UNDERWEAK IDENTIFICATION

Published online by Cambridge University Press:  25 November 2011

Xiaoxia Shi  and
Peter C.B. Phillips
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Abstract

An asymptotic theory is developed for a weaklyidentified cointegrating regression model in whichthe regressor is a nonlinear transformation of anintegrated process. Weak identification arises fromthe presence of a loading coefficient for thenonlinear function that may be close to zero. Inthat case, standard nonlinear cointegrating limittheory does not provide good approximations to thefinite-sample distributions of nonlinear leastsquares estimators, resulting in potentiallymisleading inference. A new local limit theory isdeveloped that approximates the finite-sampledistributions of the estimators uniformly wellirrespective of the strength of the identification.An important technical component of this theoryinvolves new results showing the uniform weakconvergence of sample covariances involvingnonlinear functions to mixed normal and stochasticintegral limits. Based on these asymptotics, weconstruct confidence intervals for the loadingcoefficient and the nonlinear transformationparameter and show that these confidence intervalshave correct asymptotic size. As in other cases ofnonlinear estimation with integrated processes andunlike stationary process asymptotics, theproperties of the nonlinear transformations affectthe asymptotics and, in particular, give rise toparameter dependent rates of convergence anddifferences between the limit results for integrableand asymptotically homogeneous functions.

Information

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 3 , June 2012 , pp. 509 - 547
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

Our thanks to two referees and the co-editor,Pentti Saikkonen, for helpful comments on theoriginal version. The paper originated in a 2008Yale take-home examination. The first completedraft was circulated in December 2009. Xiaoxia Shigratefully acknowledges support from the CowlesFoundation via a Carl Arvid Anderson Fellowship atYale University. Peter Phillips thanks the NSF forsupport under grant SES 06-47086 and 09-56687.

References

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